Is it permissible to communicate with a non-Muslim girl via Facebook with written messages for the purpose of inviting her to Islam, knowing that there are feelings of love towards her?

If the conversation is in a public forum accessible to others, then there is no harm in it, because that removes the causes of temptation (fitna). However, if it is in private, then it is not permissible, as it is a gateway to temptation. You must seek safety for your religion and your honor, especially since you love this girl, which Satan might exploit to lead you both to what displeases Allah Almighty.